Specialization and rigidity

TL;DR

This paper presents a general method for demonstrating that a broad class of fibers in algebraic families possess specific properties, with applications across various fields such as algebraic dynamics and Galois theory.

Contribution

It introduces a versatile method originated by Ofer Gabber for establishing properties of general fibers in algebraic families, illustrated through diverse mathematical examples.

Findings

Method successfully applied to algebraic dynamics

Provides insights into the rationality problem

Connects fiber properties with Galois and quadratic form theories

Abstract

We describe a general method, originated by Ofer Gabber, for showing that a very general fiber in a family has certain properties. We illustrate this method with concrete examples taken from algebraic dynamics, the rationality problem for algebraic varieties, Galois theory, quadratic form theory and the theory of central simple algebras.